#199223

order	512
nilpotency class	3
isoclinism	isoclinic class of #199221
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1093 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(2,6)(3,4)(7,8)(9,10)(11,15,20,30)(12,22)(13,26)(14,17)(16,24)(18,28)(19,35,23,40)(21,38,37,31)(25,34)(27,33)(29,39,32,36);(1,2,4,6,5,7,3,8)(9,11)(10,15,17,30)(12,23)(13,21)(14,20)(16,32,33,29)(18,31,26,38)(19,25)(22,40,34,35)(24,39)(27,36)(28,37);(1,3,5,4)(2,6,7,8)(9,12,14,25)(10,16,17,33)(11,19,20,23)(13,27,28,24)(15,29,30,32)(18,34,26,22)(21,39,37,36)(31,40,38,35);(10,17)(15,30)(16,33)(18,26)(22,34)(29,32)(31,38)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,e^2=hi,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ch,e^b=ei,g^b=gi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106419);